Expansion of function in polar coordinates

Question

I'd like to expand a function in polar coordinates to something that splits radius and angle

$f(r,\theta)=\sum_i A_i(r)B_i(\theta)$

I've found some hints on the internet by the name of polar Fourier transform, but I didn't find a Wikipedia page or a good explanation. What is the name for such a decomposition and how to find a basic description?

Is using Bessel function the only way to find such a factor decomposition?

Is it possible (or useful) to write this operation in complex number representation for the 2D coordinates?

Answer 1

Consider the 2D Fourier transform in rectangular coordinates,

$$ F(u,v) = \int_{-\infty}^{\infty} \left[ \int_{-\infty}^{\infty} f(x,y)e^{2i\pi ux}dx \right]e^{2i\pi vy}dy $$ One can take 1D FT in X direction then in Y direction. But for Polar coordinates the expression changes, Let , $ux+vy = \rho r \cos (\phi-\theta)$, then $$ F(u,v) = \int_{0}^{2\pi} \int_{0}^{\infty} f(r,\theta)e^{2i\pi\rho r \cos(\phi- \theta)}rdr d\theta $$ Now the function $f(r,\theta) $ has seperability in polar coordinates if it can be written in the form , $$f(r,\theta)= f_r(r)f_{\theta}(\theta)$$ Then the function $f(r,\theta)$ is a circularly symmetric with $f_{\theta}(\theta)=1$ , examples include like that of a cylinder etc. Read more about Hankel transform, 2D Polar Fourier transform and functions of this type.